DOCUMENT RESUME ED 069 506 AUTHOR Ninety … · DOCUMENT RESUME. ED 069 506 SE 015 197. AUTHOR Holland, Bill TITLE Learning Activity Package, Geometry. INSTITUTION Ninety Six High

DOCUMENT RESUME

ED 069 506 SE 015 197

AUTHOR Holland, BillTITLE Learning Activity Package, Geometry.INSTITUTION Ninety Six High School, S. C.PUB DATE 72NOTE 58p.

EDRS PRICEDESCRIPTORS

MF-$0.65 HC-$3.29Curriculum; *Geometry; *Individualized Instruction;*Instructional Materials; Mathematics Education;Objectives; *Secondary School Mathematics; TeacherDeveloped Materials; Teaching Guides; Units of Study(Subject Fields)

ABSTRACTA set of three teacher-prepared Learning Activity

Packages (LAPS) in geometry, the units cover the topics of distance,lines, planes, separation; angles and triangles; and congruences. Theunits each include a rationale for the material, a list of behavioralobjectives, a list of resources including texts (with readingassignments and problem sets specified) and tape recordings, astudent self-evaluation sheet, suggestions for advanced study, andreferences. For other documents in this series, see SE 015 193, SE015 194, SE 015 195, and SE 015 196. (DT)

L EARNING

ACT IV ITY

PACKAGE

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DISTANCE, L INES,

PLANES, SEPARAT ION

1111.=11111V

JIEIJED LAP NUMBER 26

1

WRITTEN BY Bill Holland

3

RATIONALE

DISTANCE, LINES, PLANES, SEPARATION

The SYSTEM of GEOMETRY is an historically important one and is

essentially the system as first created by Euclid. However, the parti-

cular development followed here is not truly Euclidian. You will use

your knowledge of the set of real numbers singe Geometry is introduced

as a mathematical study of sets of points, and geometric figures are

treated as such.

BASIC POSTULATES about distance are expressed in terms of one-to-one

correspondence between sets of real numbers and sets of points. The

equality and order properties of a field are used to develop concepts

of distance, betweenness, and congruence. The absolute value of a real

number is necessary for a concise and mathematically accurate statement

of how to compute the distance between two points on a number line.

The structure of a postulated system is discussed and the necessity

for using precise language is stressed! As you work with definitions,

postulates, and theorems, they will become more meaningful to you. It is

important to clarify the use of the three undefined terms - POINTS,

LINE and PLANE!!

SECTION I

Behavioral Objectives

By the completion of the prescribed course of study, you will be able

to:

1. Given any set:

a. Write a word description of the set.

b. Write a description using the roster method.

2. Given any pair of sets:

a. Determine if they are equal.

b. Determine if one set is a subset of the other.

c. Name their intersection.d. Name their union.e. Determine if they intersect.

3. Given a set of real numbers, state which elements are:

a. integersb. rational numbers

c. irrational numbers

4. Given statements, using the relations listed below, determine

whether they are TRUE or FALSE:

a. < (less than)

b. > (greater than)

c. s (less thah or equal to)

d. (greater than or equal to)

5. Given a list of statements, determine which ones are examples of

the following properties:

a. The Trichotomy Propertyb. The Transitive Property of Order

c. The Property Related to the Taking of Opposites

d. Addition for Inequalitiese. The Property Connecting Order with Equality

f. Multiplication for Inequalities

6. Decide for what replacement of x, the real number -x will name:

a. the number zerob. a positive real numberc. a negative number

7. Given a real number, decide whether or not it has a positive square

root.

8. Given a non-negative real number, name its positive square root

and its negative square root.

2

SECTION I

Behavioral Objectives (cont')

9. Given any real numbers

a. Ix1

b. ix + yc. Ix) IYI

ye. 1x1 IYI

x and y, determine the following:

10. Be able to graph algebraic sentences of the following forms:

a. Ixl < a

b. Ixl > a

c . Ixl = a

d. IxI < a

e. Ix' >_a

4

3

SECTION I

RESOURCES

I. Readings:

1. Moise: Geometry - #1, #2 pp. 1-3, 8-11, 15-17; #3 - #8 pp. 21-

24; #9 - #10 pp. 26.

2. Jurgensen: Modern Geometry - #1, #2 pp. 1-3, 5-7, 8-10; #3 - #8

pp. 11-12, 14-17; #9 - #10 pp. 18-20.

3. Nichols: Modern Geometa- #1, #2 pp. 1-8; #3 - #8 pp. 16-21(objectives 6, 7, and 8 not covered); #9, #10 p. 22 (objective 10not covered).

4. Anderson: School Mathematics Geometry - #1, #2 pp. 17-19, 21-23;#3 - #8 pp. 25-28, 29-30; #9, #10 pp. 29-31 (objective 10 not

covered).

5. Lewis: Geometry - #1, #2 pp. 10-11; #3 - #10

II. Problems:

1. Moise: Geometry - #1, #2 pp. 4-8 ex. 1-3, 6-13, pp. 12-13 ex. 1,3-5, pp. 18-20 ex. 1-12, 14-15, p. 50 ex. 1-2; #3 - #8 pp. 24-25ex. 1-9, pp. 50-51 ex. 3, 5, 12; #9, #10 p. 27 ex. 1-6, p. 50 ex.

4

2. Jurgensen: Modern Geometry - #1, #2 pp. 3-5 ex. 1-36, pp. 7-8ex. 1-32 pp. 10-11 ex. 1-16 p. 45 ex. 1-10, p. 47 ex. 1-15; #3 -

#8 pp. 12-13 ex. 1-36, p. 17 ex. 1-36, pp. 47-48 ex. 16-20; #9,#10 p. 46 ex. 19-22, p. 47 ex. 16-20, 27-29, 35, 37, 39, 45, pp.20-21 ex. 1-40.

3. Nichols: Modern Geometry - #1, #2 p. 8 ex. 1-6; #3 - #8 pp. 18-19 ex. 1 (a-1), 10, pp. 20-21 ex. 1(a-1), 6, 8 (a-h), 10; #9,#10 p. 22 ex. 1-3.

4. Anderson: School Mathematics Geometry - #1, #2 pp. 20-21 ex. 1-13, pp. 23-24 ex. 1-9; #3 - #8 pp. 28-29 ex. 1-11, pp. 32-33 ex.1-14, 26-28; #9, #10 pp. 32-33 ex. 15-25, 29-31.

5. Lewis: Geometry - #1, #2 pp. 11-13 ex. 1-12; #3 - #10

4

SELF-EVALUATION I

I. Match the statement in column A with its equivalent statement in column B.

COLUMN A COLUMN B

I. set of numbers which are a. 0

exponents in the expression:

5x7 + 4x5 + 7x3

2. set of numbers which arecoefficients in the aboveexpression

3. {Sun., Mon., Wed., Fri., Sat.,}

b. the days of the weekwhose first letters arenot "T"

c. {3, 5, 7}

d. the set of even integersless than nine and greaterthan one

4. the set of odd numbers whosesquares are even numbers e. {4, 5, 7}

5. {2, 4, 6, 8}

2. Answer the following true or false:

a. The subsets of {a} are {a} and 0.

b. Sets A and B are equal if A is the set of all whole numbers3 22

between 7 and 3 and B is the set whose elements are

2, 3, 4, 5, 6, 7.

c. The union of the set of primes and the set of positiveeven integers is the set of all positive integers.

d. It is possible for the intersection of a line and a planeto be the empty set.

e. If two lines intersect, then their intersection is neverthe empty set.

f. The intersection of the set of real numbers and the setof integers is the set of integers.

3. Given A = {-3, 0, 0.25, r, 7, 3, 0.82, 0.763..., -1}

a) Which of the elements of A are rational numbers?

b) Which of the elements of A are irrational numbers?

c) Which element in A is greater than each of the other elements?

5

SELF-EVALUATION I (cont')

d) Which element in A is smaller than each of the others?

e) Between which two integers does I lie?

f) Between which two integers does it lie?

4. Write the following, using the symbols of order, <, s, >, a.

a) x is a number greater than O.

b) y is a number between -1 and 2.

c) w is a number between -2 and 1 inclusive.

d) p is a positive number.

e) n is a negative number,

f) n is a non-negative number.

5. Matching. Select a letter of the name of the property in column Bwhich matches the number of the example of the property iiicolumn A. (You may use the property more than once.)

COLUMN A

a) If r< s and s< t, then r< t

b) If r < s and a > 0, then ar < as

c) If r < s, then r + y < s + y.

d) If r < s, then -3r > -3s.

e) If r # 3 and r )( 3, then r > 3

f) If r + 3 = s, then r < s.

g) If r < 3, ther -r > -3.

COLUMN B

1) Trichotomy Property

2) Transitive Property of Order

3) Property Related to theTaking of Opposites

4) Addition for Inequalities

5) Property Connecting Orderwith Equality

6) Multiplication for Inequalities

6. Write the simplest real number name for each of the following:

a) lx1

b)42:

6

c) a squarri root of 81

d) $4:4"

e) 101

f) rT1

SELF-EVALUATION I (Cont')

g) 1131 - 1171

h) 113 - 171

i) 1-131 + 1171

7. Graph the following algebraic statements (geometric interpretation):

a) :x1 .4- 3

b) 1x1 > 3

c) 1x1 < 3

d) 1x1 ?. 3

TEST.IF YOU HAVE PLASTERED THE BEHAVIORAL OBJECTIVES, TAKE YOUR PROGRESS

37

ADVANCED STUDY I

1. Determine the solution sets of the following:

a) Ix - 3.1 s6

b) I5x - 91> 2

c) X2 - 6x + 9 < 5

d) x2 -11x + 10 > 1

el x+ 1 <

' x - 3

.F1 x + 7si 2x - 6

3

9 8

SECTION II


By the completion of the prescribed course of study, you will beable to:

11. Decide whether or not given statements are true or false whenthe statements involve the following Theorems, Definitions,and Postulates:

a. Distance Postulateb. Ruler Postulatec. Definition of a Coordinate Systemd. Ruler Placement Postulatee. Definition of Betweenf. Line Postulateg. Definition of a Segmenth. Definition of Length of a Segmenti. Definition of a Rayj. Definition of Opposite Raysk. The Point-Plotting Theorem1. Definition of Midpoint of a Segmentm. Definition of a Bisectorn. Theorem: Every segment has exactly one midpointo. Definition of Absolute Valuep. Definition of "vim"

12. Given the coordinates of two points, compute the distance betweenthe two points.

13. Given the coordinates of two points, compute the coordinate ci.the midpoint of the segment.

14. Given a coordinate system on a line with the coordinates of twopoints stated, apply the Ruler Placement Postulate and assign 0to one of the points and determine the positive coordinate ofthe other point.

15. Decide whether or not given statements are true or false when .

they involve the following Theorems, Postulates, and Definitions:

a. Definition of Spaceb. Definition of C011inear and Coplanarc. Theorem: If two different lines intersect, their intersection

contains only one poitn.d. Postulate: If two points of a line lie in a plane, then the

line lies in the same plane.e. Theorem: If a line intersects a plane not containing it,

then the intersection contains only one point.f. The Plane Postulateg. Theorem: Given a line and a point not on the line,.there is

exactly one plane containing them.h. Theorem: Given two intersecting lines, there is exactly one

plane containing both.

10 9

SECTION II

Behavioral Objectives (cont')

i. Postulate: If two different planes intersect, then theirintersection is a line.

j. Definition of a Convex Setk. Definition of Separation1. Plane and Space Separation Postulatesm. Definitions of Half-line, Half-plane, and Half-space

16. Given a drawing showing Points, Lines.; Planes, and a list ofSets, determine which of the sets contain members which are:

a. Collinearb. Coplanarc. In the same Half-spaced. In the opposite Half-plane

17. Given any pair of geometric figures (Points, Lines, Planes),name their intersection.

18. Determine from a given set of drawings which ones represent convexsets.

RESOURCES II

I. Readings:

1. Moise: Geometry - #11-#14 pp. 31, 33-35, 38-39, 41-43, 44-45,46-48; #15-#18 pp. 55-58, 59-61, 63-65.

2. Jurgensen: Modern Geometry - #11-#14 pp. 14-16, 18-19, 22, 26-28;#15-#18 pp. 22-23, 26, 31.

3. Nichols: Modern Geometry - #11-#14 pp. 22-25, 26-27, 28-29, 32-34;#15-#18 pp. 30-31, 40.

4. Anderson: School Mathematics Geometry - #11-#14 pp. 43-46, 49-52,54-57; #15-#18 pp. 69-70, 74-76, 84-86, 88.

5. Lewis: Geometry - #11-#14 pp. 13-17, 22-26, 37-39; #15-#18

II. Problems:

1. noise: Geometry - #11-#14 pp. 31-32 ex. 1-5, pp. 35-37 ex. 1-11,pp. 39-40 ex. 1-7, pp. 43-44 ex. 1-4, 6-14, pp. 45-46 ex. 1-7,pp. 48-49 ex. 1-5; #15-#18 pp. 58-59 ex. 1-7, pp. 61-62 ex. 1-12,pp. 66-67 ex. 1-15.

2. Jurgensen: Modern Geometry - #11-#14 pp. 20-21 ex. 1-18, pp. 23-24 ex. 1-12, pp. 29-30 ex. 1-12, 14-15, 19-20, 23-24; #15-#18 pp.25-26 ex. 13, 16-18, 21-22, 25-32.

3. Nichols: Modern Geometry - #11-#14 pp. 25-26 ex. 1-11, p. 27 ex.1-8, pp. 29-30 ex. 1-9, pp. 34-35 ex. 1-7; #15-#18 pp. 31-32 ex.1-5, p. 41 ex. 8-12.

4. Anderson: School Mathematics Geometry - #11-#14 pp. 47-49 ex.1-10, pp. 53-54 ex. 1-14, pp. 58-59 ex. 1-5; #15-#18 pp. 72-74ex. 1-13, p. 77 ex. 1-10, pp. 87-88 ex. 1-11, pp. 90-91 ex. 1-10.

5. Lewis: Geometry - #11-#14 pp. 18-20 ex. 1-10, pp. 26-27 ex. 1-6,pp. 39-40 ex. 1, 3, 5, 7, 9.

III. Audio:

Objectives 15-18: Wollensak Teaching Tape R-3605

11

SELF-EVALUATION II

1. Answer the following True or False:

a. The distance between any two points can never be anegative real number.

b. Every point on a number line corresponds to a rationalnumber.

c. The number assigned to a point is called the coordinateof the point.

d. If PQ = 0, then P and Q are two names for the same point.

e. The distance between two points is the absolute valueof the difference of their coordinates.

f. If AB + BC = AC and A, B, and C are different pointson the same line, then B is between A and C.

g. Given a coordinate system on a line, if the coordinateof point R is -7 and the coordinate of point S is 3,then RS = 4.

h. Given a coordinate system on a line, if the coordinateof point A is 8, the coordinate of point B is 20, andC is the midpoint of AN, then the coordinate of C is 12.

i. If CA and 5 are opposite rays, then C is between A andB.

j. If A and B are two different points, then AB must namea positive real number.

2. Complete each statement with the correct sYmbol or word:

a) A, B, and C are three points on a line. If AC = 7 and BC = 5, and

B is between A and C, then AB =

b) If 2 is the coordinate of A and -7 is the coordinate of B, then

AB =

3. a) C, the midpoint of WIT, has coordinate 5. If the coordinate of A isgreater than the coordinate of B, and BC = 9, what are the coordinatesof A and B?

b) If A is -3 and C is 11, then the midpoint B corresponds to

SELF-EVALUATION II (Cont')

4. Given a coordinate system on line RS. If the Ruler Postulate isapplied with R corresponding to 0, what is the new coordinate of Swhen:

a) R is -7 and S is 6.

b) R is 3 and S is 11.

c) R is -5 and S is y.

5. Decide whether or not the following statements are true or false.

a. A line and a plane always have a point in common.

b. Two lines always lie on the same plane.

c. There are lines which do not intersect each other.

d. If three points are collinear, they are coplanar.

e. A point and a line always lie in one and only oneplane.

f. Given two different points A and B, there are atleast two different lines that contain both A and B.

g. A line has two end points.

h. Given two points, there is more than one planecontaining them.

i. Given two points A and B, if A E line L1 and B e line

L2 and AB 5 L , then L1 = L2.

j. There exists three lines in a plane such that theyseparate the plane into seven regions.

k. Given: Points A, B, and C lie in plane P. Points

A, B, and C lie in plane R. You can conclude thatplane F is the same plane as plane R.

1. A circle is a convex set.

m. Two half-planes with a common edge are coplanar.

n. A half-space is a convex set of points.

o. We have a postulate stating that two intersecting linesdetermine a plane.

p. There are three conditions that determine a plane.

q. Given a line L, if L c plane R and L plane S, then R = s.

13

SELF-EVALUATION II (cont')

6. Study the three dimensional figure given (in which R, S, T, U arecoplanar) and answer the following questions:

a) Are R, S, T, U, and X coplanar?

b) Are V, U, and W collinear?

c) Are V, T, X, and R coplanar?

d) Are V, T, S, and W coplanar?

e) Are V, S, and W coplanar?

f) Are V, U, R, and W coplanar?

g) Does the interior of the figure represent a convex set of points?

7. Write a word or phrase which would make each of the following a cmpleteand Niue statement. In the figure below, plane E contains lines AB, GK,

and , all of which intersect at M.

/E

a) According to the pastulatei_since T and R are indifferent half-planes formed by AB, then TR must

b) Points A and B are said to lie on sides of GK.

c) Points A and G lie on the same side of tl because because

d) According to one of our Postulates, if we know only that the givenplane contains A and

ADVANCED STUDY II

1. An airplane is 4 miles east and 2 miles north

of Baltimore. Five minutes later it is 15 miles

east and 2 miles north of Baltimore. How far

and how fast did it travel in the 5 minutes?

2. Airplane B is 4 miles east and 2 miles north of

Baltimore. In four minutes the plane traveled

12 miles due east. How far east and how far

north is the plane from Baltimore now? What

was its rate of speed?

3. A boat is traveling from buoy #4 to buoy #6.

Buoy #4 is located 6 miles east and 2 miles

south of a lighthouse. Buoy #6 is located 16

miles east and 4 miles south of the lighthouse.

When the boat completes half of its trip, how

far south of the lighthouse will it be?

15

REFERENCES

I. Textbooks:

1. Moise, Downs: Geometry (Addison-WesleyPublishing Co., Inc., 1964).

2. Jurgensen, Donnelly, Dolciani: ModernGeometry (Houghton-Mifflin Co., WgT

3. Nichols, Palmer, Schacht: Modern Geometry(Holt, Rinehart, and Winston, Inc., 1968).

4. Anderson, Garon, Gremillion: School Mathe-matics Geometry (Houghton-Mifflin Co., 1968).

5. Lewis: Geometry, A Contemporary Course (D.Van Nostrand Co., Inc., 1968).

II. Audio:

1. Wollensak Teaching Tapes: R-3605

1EARNING

ACT I.V I TY

PACKAGE

ANGLES AND TRIANGLES

WRITTEN BY Bill Holland

RATIONALE

ANGLES AND TRIANGLES

We are sure that most of you can draw pictures of angles and triai.71(.,?

In science we refer to angles in relation to refraction and reflection of

light, to the bonding of angles in molecular structure, and to triangulaHon

in determing distance to a star. But, if we were to ask you to give a pre-

cise definition of angles and triangles, you might have some difficulty. In

this section we will discuss the basic definitions of these geometric figures.

The idesa we thus ,:ammunicate to you will be used as a basis for proving some

theorems later on in this LAP.

Many of the properties of triangles and angles are dependent on the

measures of these geometric figures. We will discuss measurement of angles,

including how to measure angles and what postulates allow us to measure them.

We will also refer to some of the special relationships, based upon measure,

which exist between angles.

You will find that we can classify any angle into one of three catennr-

ies. There will be no new postulates in this section, but we will be usino

pur previous definitions and postulates to prove a few theorems about anales.

It is not necessary for you to memorize the proofs of these theorems, but it

is recommended that you study them and try to understand the thinking bch4r1

the proofs, and become aware of how to develop your own simple proofs.

191

Behavioral Objectives:

to:

SECTION I

By the completion of the prescribed course of study, you will be abL

I. Given a drawing of an angle, properly labeled,

a. Name the vertex of the angle.

b. Name the sides of the angle.

c. Name the angle.

2. Given a drawing of a triangle, properly labeled,

a. Name the sides cf the triangle.

b. 'game the angles of the triangle.

c. Name the triangle.

d. Identify an included side.

e. identify an included angle.

f. Identify a side opposite a given anRle.

g. Identify an angle opposite a given side.

3. Given a drawing of an angle, properly labeled, and any point,decide whether the point lies in the interior of the angle, inthe exterior of the angle, or on the angle itself.

4. Given a drawing of a triangle, properly labeled, and a word des-cription of a point, decide if the point lies in the interior ofthe triangle, the exterior of the triangle, or on the triangleitself.

5. Answer correctly true-false, matching, or comple*:ion type ques-tions involving:

a. The definition of an angle.

b. The definition of a triangle.

c. The definition of the interior of an angle.

d. The definition of the exterior of an angle.

2

SECTION I - Behavioral Objectives (cont')

e. The definition of the interior of a triangle.

f. The definition of the exterior of a triangle.

6. Given a drawing of one or more angles, estimate the measure ofthe angles to the nearest 10 degrees.

7. Given a composite drawing of angles, classify each of the anglesas acute, right, or obtuse.

8. Given a drawing of a pair of lines, rays, or segments; determinewhether or not the pair of geometric figures are perpendicular.

9. Given a measure of an angle, compute the measure of its supple-ment.

10. Given a relationship between two supplementary angles, computeThe measure of the given angles through the use of the definition3f supplementary angles and the supplement postulate.

11.. Given the measure of angle, compute the measure of its comple-ment.

12. Give. a relationship between two complementary angles, computethe measure of the given angles through the use of the definitionof compinmentary angles.

13. Answer currectly true-false, matching, or completion type questionsinvolving:

a. The angle Measurement Postulate.

b. The definition of the measure of an angle.

c. The Angle Construction Postulate.

d. The Angle Addition Postulate.

e. The definition of a linear pair.

f. The definition of supplementary angles.

g. The Supplement Postulate.

14. Answer correctly true-false matching or completion type questionsinvolving:

a. Acute angles

b. Obtuse angles

c. Right angles

3

SCCTiON I - Behavioral Objectives (cont')

d. The definition of a ray, a segment, or perpendicular lines.

e. The definition of complementary angles.

f. The definition of vertical angles.

g. The definition of adjacant angles.

h. Postulates and theorems relating to angles and their relation-ship to other angles.

15. Given a drawing of intersecting lines and rays, and a list ofangle relationships; match a pair of angles with the correct rela-tionship involving:

a. Acute angles

b. Obtuse angles

c. Right angles

d. Equal angles

e. Supplementary angles

f. Complementary angles

g. Linear pair

h. Vertical angles

i. Congruent angles

j. Adjacent angles

SECTION I

Resources

I. Readings:

1. Moise: Geometry - #1 - #5 pp. 75-77; #6 - #8 pp. 80-82, 87-88; #9 -.#12 pp. 83, 87; #13 - #15 pp. 82-83, 88-92.

2. Jurgensen: Modern Geometry - #1 - 5 pp. 30-31; #6- #8 pp.33-36, 39, 125; #9 - #12 pp. 40, 130-131, 133; #13 - #15 pp.40, 125, 131, 138-139.

3. Nichols:. Modern Geometry - #1 - #5 pp. 38-40, 42-43, 47-48;#6 - #8 pp. 111-115, 121-122; 49 - #12 pp. 49-50, 59-60; #13 - #15 pp. 118-121, 125-127.

4. Anderson: School Mathematics Geometry - #1 - #5 pp. 99-104,107-108; 45 - #8 pp. 111-115, 121-122; #9 - #12 pp. 115, 121-122; #13 -- #15 pp. 18-121, 125-127.

5. Lewis: Geometry - #1 - #5 pp. 17-18; #6 - #8 pp. 27-31, 33-34; #9 - #12 pp. 31-32; #13 - #15 pp. 59-61, 63.

II. Problems:

1. Moise: Geometry.- #1 - #5 pp. 78-79 ex. 1 - 18; #6 - #8 pp..34-8.; ex. 6-10, pp. 38-89 ex 1, 3, 5; #9 - #12 p. 84 ex. 1-5,pp. 85 -S6 ex. 13, 15-18, pp. 88-89 ex. 1-2, 6-8; #13 - #15pp. 85-87 ex. 12, 14, 20, p. 94 ex. 1-8.

2. Jurgensen: Modern Geometry - #1 - #5 p. 32 ex. 1-12; #6 - #8p. 38 ex. 1-22, p. 129 ex. 1-20, p. 141 ex. 1-10; #9 - #12pp. 42-43 ex. 1 -34. p. 135 ex. 1-10, p. 141 ex. 11-19, p. 143ex. 39-42; #13 - #15 p. 129 ex. 1-20, pp. 135-136 ex. 11-14,19-22, p. 141 ex. 1-14.

3. Nichols: Modern Geometry - #1 - #5 p. 41 ex. 1-6, pp. 43-44ex. 1-10, pp. 48-49 ex. 1-4; 46 - #8 p. 46 ex. 1-3, p. 58 ex.1-6; #9 - #12 pp. 50-51 ex. 1-8, p. 60 ex. 1-9, pp. 78-79 ex.4-5, 8-12; #13 - #15 pp. 53-54 ex. 1-10, p. 79 ex. 7.

4. Anderson: School Mathematics Geometry - #1 - #5 pp. 106-107ex. 1-8, p. 110 ex. 1- 12;' #6 - #8 pp. 116-117 ex. 1-8, pp.123-124 ex. 1-3, 10-11; #9 - #12 p. 118 ex. 9-15, p. 123 ex.

5-7; 413 - #15 p. 138 ex. 1-2, 6, 11.

5. Lewis: Geometry - #1 - #5 p. 19 ex. 3-5; #6 - #8 pp. 32-33ex. 1-3, 8-10, pp. 34-35 ex. 1-8; #9 - #12 p. 33 ex. 4-7,11; #13 - 415 p. 61 ex. 1-2, p. 64 ex. 4-8, pp. 67-68 ex. 3-4, 6, 8, 10.

SELF-EVALUATION I

1. Which of the following is not a name for LQ in the illustration?

a) LPQR

b) LTQS

c) LQSR

d) LTQP

2. In the figure at the right, A, B, and C are collinear:

a) Name five angles

4

4

b) What is the vertex of all the angles named?

3. Using the given illustration, complete the following:c

A

a. Name the triangle.

b. Name the angles of the triangle.

c. The side opposite LA is .

d. The angle included by sides Wand ET is .


e. The side included by LA and LC is

f. The angle opposite AC is

4. For the following diagram, classify each of the following statementsas True or False.

p

1.

a. Z is in the interior of LBED.

b. P is in the interior of LBED.

c. P belongs to LBED.

d. A belongs tl LAED.

e. Z is in the exterior of LAED.

f. B belongs to LAEC.

g. E belongs to all four angles illustrated.

h. D is in the exterior of LAEC.

5. Use the following illustration1rand classify each statement true or false:

Ca. If P is in the same half-plane will edge AC as T, andin the same half-plane with edge CT as A, then P is

in interior of LC.

b. If P is located as described in (a), then P is inthe interior of aTCA.

7

SELF-EVALUATION I (cont1)

c. If P is in the interior of both LC and LA, then Pis in the interior of ATCA.

6. Complete the following statements by selecting the one best word orphrase to make each a true statement:

A. An angle is the of two which have the

end-point and do not lie in the same

B. If A, B, and C are any points, then the

of segments AN, AC, and BC is a triangle.

C. The of LBAC is the set of all points in plane E that

do not lie in the interior of LBAC or on the

D. The interior of an angle is the of two

E. A point lies in the interior of a triangle if it lies in the

of each of the triangle.

F. Which of the following is not a convex set?

(1) tRST (2) half-plane (3) g (4) interior of AMAD

7. Matching: Estimate the size of the angles without using a protractor.Decide which of the angles shown have measures within the indicatedranges listed at the right.

(a)

(b)

8

(1) 15 < x < 35

(2) 70 < x < 90

(3) 80 < x < 100

(4) 45 < x <60

(5) 90 < x < 110

(c)

(d)


8. Matching: Use the illustration below and correctly match the items ofcolumn B with items in column A.

COLUMN A

a) LDGF

b) GAGE and LOGB

c) LBGA

d) LEGC

e) LEGF and LFGA

279

COLUMN B

1) acute angle

2) obtuse angle

3) right angle

4) vertical angles

5) none of the above

9.


Using the diagram decide which of the following statements is true orfalse.

(a) If PE .L CO, then m LCPA + m LOPE.

(b) aA is perpendicular to 5.

(c) CD is perpendicular totl.

(d) PE is perpendicular to PC.

(e) t is perpendicular to AB.

10. Determine the measure of the supplement of an angle whose measure is:

(a) 75

(b) 68

(c) 35.5

(d) m + y

(e) 180 - x

(f) 90 - y

11. a) Twice the measure of an angle is 30 less than five times the measureof its supplement. What is the measure of both angles?

10


b) Two angles are supplementary. The measure of one is 24 less thantwice the measure of the other. What is the measure of the largerangle?

12. Determine the measure of the complement of an angle whose measure is:

(a) 40

(b) 68

(c) 46.5

(d) r:

) 45 + n

13. a) Given m LR + m LX = 90 and m LS + m LY = 90. If m LR > m LS,

which of the following is true?

(1) m LX < m LY

(2) m LX = 90 - m LY

(3) m LX > m Ly

(4) m LX + m LY = 90

b) The measure of an angle is 7 times the measure of its complement.Find the measure of the larger angle.

11

99


14. Using the illustration select the item from Column B thatcorrectly relates to th ;Otement in Column A;

\

(S

1.

COLUMN A COLUMN B

a) m LHPZ + m LWPZ = m LHPW 1) Angle Measurement Postulate

b) m LHPZ + m LSPR = 180 2) Angle Construction Postulate

c) m LHPW 180 3) Angle Addition Postubte

d) tRPQ and L.QPH form a linear pair 4) Definition of a Linear Pair

e) The measure of LSPQ is m LSPQ 5) Definition of the Measure ofan Angle

6) Definition of SupplementaryAngles

7) Supplement Postulate

15. If you were given that La is the complement of Ly, Lb is tl.! comple-ment of Lx and Lx g Ly, what postulate or theorem would you use toprove La g-Lb?

b) Answer True or False:

1) If P4 and Pit form a right angle, then they areperpendicular and intersect at P.

2) If two lines intersect, then any two adjacent anglesform a linear pair.

3) The complement of an acute angle is obtuse.

4) If two intersecting lines form one right angle, thenthey form four right angles.

30 12


5) Vertical angles are always right angles or acute angles.

6) Two angles are vertical angles'if their sides form two

pairs of opposite rays

16. Use the given figure to name the required angle or angles:

Q

a) Name a pair of equal acute angles.

b) Name a pair of vertical angles which are obtuse,

c) Name a pair of equal angles which form a linear pair.

d) Name a pair of adjacent complementary angles.

e) Name an angle which is a supplement to /JOS.

IF YOU HAVE MASTERED THE OBJECTIVES, TAKE YOUR PROGRESS TEST.

13

31

Objective (6)

Objective (6)

ADVANCE STUDY

Section I

APPLICATIONS

Arsle of Incidence - the angle between'a li0 b ray (incident ray)

and the perpendicular (normal) to the point where it meets the

reflecting surface (point of incidence).

Angle of Reflection - the angle between the normal and the reflected

light ray.

law of Reflection - a ray of light striking a reflective enlace will

be reflected eo that the Engle of refleqgma is equal to the

angle of incidence. In optics, all angles are measured from

the normal.

(1) In the Pigure I below, will the light ray be reflected

to point P, S, or T ?

Normal

TP

S

Figure I

(2) Which of the points in Figure II will not receive light

from either ray A or ray B ?

A

32

Pisure II

14

ADVANCE STUDY I (cont')

.Objective (9) (3) In the water molecule, the two hydrogen atoms are on each side

of the oxygen atom and form an angle of 1050 pith the oxygen

stomas shown below.

The supplcment of this angle indicates the extent to which

the molecule departs from being linear. That is the angle

for water ?

Objective (14) (4) Three billiard balls are located as shown in the diagram below.

Which point will ball A have to hit both ball B and ball C ?

cue stick

(Cont' on following page)


5. a) Given AABC. A-D-B, B-E-C, C-D-F, and D-G-E.

1) Is G in the interior or exterior of AABC?

2) Does gt intersect At?

3) G and F are on opposite sides of

b) Twice the measure of an angle is 30 less than five times the measureof the supplement. What is the measure of the angle?

c) If the sum of the supplement of an angle and three times the com-plement equals 250°, how large is the angle?

d) If two straight lines intersect so that one of the smaller anglesformed equals 30°, how large is each of the other angles?

6. a) If, in a plane, m LBAD = 65 and m LDAC = 32, what is m SCAB?

b) In half plane H, BA and BE are opposite rays, LABG 5 LKBG, andLKBD g OBE. Find m LGBD.

16


) If OA, OB, and OC are three distinct rays in a plane such that notwo of them are opposite, is each of the following statements trueor false? (Recall that just one exception will make the statementfalse).

(1) m LAOB m LBOC = m LAOC

(2) m LAOS m LBOC = m LAOC = 360

If you write that one of the statements is false, justify youranswer.

r'oro 17

SECTION II


By the completion of the prescribed course of study, you will be ableto:

16. Given a statement, rewrite the statement in the hypothesis-conclusion form. (in a conditional sentence)

17. Given a statement, complete a proof of the statement in twocolumn form using any of the following:

a) vertical angles are congruent

b) definition of between

c) Angle Addition Postulate

d) supplement of congruent angles are congruent

e) complement of congruent angles are congruent

Resources

I. Readings:

1. Moise: Geometry - #16 - #17 pp. 95-98.

2. Jurgensen: Modern Geometry - #16 - #17 p. 91.

3. Nichols: Modern Geometry - #16 - #17 pp. 76-77.

4. Anderson: School Mathematics Geometry - #16 - #17 pp. 82-83.

II. Problems:

1. Moise: Geometry - #16 - #17 p. 96 ex. 1-2, pp. 98-99 ex. 1-8.

2. Jurgensen: Modern Geometry - #16 - #17 pp. 92-93 ex. 1-16,p. 130 ex. 21-28, p. 136 ex. 23-32, pp. 142-143 ex. 21-37.

3. Nichols: Modern Geometry - #16 - #17 p. 78 ex. 1-3.

4. Anderson: School Mathematics Geometry - #16 - #17 pp. 83-84ex. 1-18, p. 124 ex. 8-9, 12, p. 129 ex. 7-10, 12.

18

SELF-EVALUATION II

1. Rewrite each of the following below in "If then

a) Supplements of congruent angles are congruent.

b) The intersection of two planes is a line.

c) Vertical angles are congruent.

d) Congruit angles have the same measure.

e) Two angles with the same measure are congruent.

2. Complete the proof by filling in the blanks:

Given: The figure withm LABC = 90 = m LDBE

Prove: LABD 5 LCBE

" form:

PROOF: Statements REASONS

1. m LABC ='90 = m LDBE 1. Given

2. LABD is complementary to LDBC 2.

3. 3. Definition of Complementary Angles

4. LDBC 4 LDBC 4.

5. 5.

19

SELF-EVALUATION II (coat')

3. Complete the proof by filling in the blanks:

Given:

Prove:

The figure with GA opposite to

Tt and nLAGB is complementary to LEGC.

PROOF: Statements REASONS

1.

2.

3.

4. GB GC

5. m LBGC = 90

6. m LBGE = m LEGC + 90

7. m LAGB + m LEGG '.- 90 = 180

8. m LAGB + m LEGC = 90

9. LAGB is complementary to LEGC

-+ -+

GA is opposite to GE 1.

LAGB is supplementary to LBGE 2.

m LAGB + m LBGE = 180 J.

4. Supply a

Given:Prove:

4.

5.

6.

7.

8.

9.

complete proof of the following:

WY = XZWX = YZ

20

X


5. Supply a complete proof of the following:

Given: MO MN

NP 1 MN0,1,,

Ll = L2

PROVE: L3 = L4

a

I'

tyl N

IF YOU HAVE MASTERED ALL THE BEHAVIORAL OBJECTIVES, A PROGRESS TESTIS SCHEDULED. AFTER THE PROGRESS TEST, A LAP TEST IS SCHEDULED.

21

39

ADVANCE STUDY II

1. Write a complete proof for the following:

Given: NP bisects LMNO

RNS NP

Prove: L3 F. L4

2. Write a complete proof for the folltming:

Given: in 4ABC, Tr3 = BC;

D bisects AN; E bisects

1:15.

Prove: 10 = CE

3. Write a complete proof for the following:

Given: LMNP E LNPQ

N bisects LMNP4.

PS bisects LNPQ

Prove: Li a L2

. 40

22

RATIONALE

In this section we explore the concepts of congruence,

particularly the congruence relation in the g triangles.

Classical theorems about congruent triangles, tlkie bisectors,

overlapping triangles and isosceles triangles virn I intro-

duced.

It is most important that you should develop some ability.

in proving theorems. Elementary congruence theorems offer an

ideal setting for moving from geometric intuition to a certain

rigor. You will be introduced to a variety of important as-

pects of proof and should become wary of "circular reasoning".

The purpose of a proof is essentially to convince some-

one of the validity of the mathematical proposition under con-

sideration. Our primary concern will be with the nrcianization

of mathematical proofs by a deductive system.

You will find that the skills gained in analyzing and

prOVing mathematical propositions will re-enforce your ability

to objectively evaluate procedures necessary to a scientific

experiment.

1

SECTION I


By the completion of the proscribed course of study, youshould be able to:

1.. Given any pair of geometric figures-,, write a correspondence

between the vertices which is a congruence.

2. Given any pair of congruent triangles, write a correspondence

which is a congruence: -

a. between trianglesb. between pairs of segmentsc. between pairs of angles.

3. Answer correctly true-false, multiple choice or matching type

questions relating to:

a, identity congruencesb. congruent angles and equal anglesc. congruent segments and equal segmentsd. included sides of a trianglee. included angles of a trianglef. isosceles trianglesg. equilateral trianglesh. angle bisectors .

1. bisectors of an angle cf e trianglej. medians of a trianglek. altitudes of a triangle1. squares and rectangles

4. Given a drawing of one or more triangles, and the necessary infor-

mation, name:

a. the angles included by two sidesb. the side included bytwo angles

5. Given any pair of congruent triangles, name the congruence

between these triangles by applying the correct postulate:

a. S.A.S. Postulateb. A.S.A. Postulatec. S.S.S. Postulate

6. Given a suitable statement, or pprcor:. Ely :.:1..t ;d figure, write

the hypothesis and conclusion as "O. :en" and "to prove,

2 4 4

SECTION I

Resources

I. Readings:

1. Moise: #1 pp. 105-107; #2-#4 pp. 112-115, 143-145; #5-#6 pp.

119-121, 128-129.

2. Jurgensen: #1 p. 191; #2- #4 pp. 41, 59, 65, 190-191, 213-214;

#5-#6 pp. 193-195, 198-199, 91-92.

3. Anderson: #1 pp. 145-147; #2-#4 pp. 151-153, 178-179, 310-311;

#5-#6 pp. 154-160, 168-171.

4. Lewis: #1 pp. 110-116; #2-#4 pp. 118-123; #5-P6 pp. 123, 153-

155, 164-165.

5. Nichols: #1 pp. 144-146; #2#4 pp. 148-149, 163-164, 168, 236-

238, 257, 295; #5-#6 pp. 150-151, 154-155.

II. Problems:

1. Moise: #1 pp. 108-110 ex. 1-9; #2-#4 pp. 115117 ex. 1-12;

p. 145 ex. 1; #5-#6 pp. 121-122 ex. 1-2, p. 129 ex. 1-2.

2. Jurgensen: #1-#4 pp. 59-60 ex. 1, 9, p. 65 ex. 1-4, pp. 192-193

ex. 1-14, p. 195 ex. 1, 2. 4; #5-#6 p. 195 ex. 5, 6, 9-11, 13-14,

16, p. 92 ex. 1-10.

3. Anderson: #1 pp. 148-151 ex. 1-10; #2-#4 pp. 154-155 ex. 1-4,

7-13, p. 157 ex. 2, p. 179 ex. 1-4; #5-#6 p. 161 ex. 1.

4. Lewis: #1 pp. 116-117 ex. 1-8; #2-#4 #5-#6 pp. 123-125

ex. 1-10, pp: 125-126 ex. 1-12.

5. Nichols: #1 pp. 146-147 ex. 1-5; #2-44 ; #5-#6 p. 152 ex. 1,

p. 156 ex. 1.

3

45

SELF-EVALUATION I

1. For each of the figures, complete the correspondence between thevertices which is a congruence for the pairs of congruent figuresas indicated:

A.

B.

C.

D.

E.

CAB 4-+ ?

Q0R-(-+?

NOLM ?

MAD --+ ?

SELF-EVALUATION I (conr')

2. For each figure given below, list all the u.rrespondences betweenthe vertices for pairs of congruent triangle::

a) 'a)

3. Below are listed six pairs of corresponding parts of two congruenttriangles. Name the congruent triangles:

MK

1314 KF

AW = iF

LA = LM

LB = LK

LW = LF

4. Given APQN = ARXW. Name the six pairs of corresponding parts.

5. Answer the following as true or false, if False, tell why.

a) In AJFK, LK is included between Fuld F.

b) Every angle has exactly one bisactor.

c) Two segments are equal if they have the same length.

d) If D is in the interior of LTAB, then A-6 bisects LTAB.

e) Every equilateral triangle is congruent.

f) A triangle is congruent to itself.

g) In APSE, 15.-§ is included by LP and LT.

h) The median of a triangle bisect!, 4nd angle of the triangle.

i) All isosceles triangles are equilateral,

j) If AADM = ABEN, then R = err and LD LN.

SELF-EVALUATION i (Cunt')

k) All rectangles are squares.

I) If Alijs a median of AABC, thei 13D = DC.

6. Given the figure as shown. List

the part which correctly fillsthe blank.

a) side, angle, side of AACD:

b) angle, side, angle of AABC: ,

c) side, angle, side of APAD: , LDPA,

d) angle, side, angle of ACDP: LCDP, LCPD

e) angle, side, angle of ADAQ: LDAQ, LDQA

7. If for each of the pairs of triangles sketched below, like markingsindicate congruent parts, name the congruence postulate which willprove the triangles congruent. If none, write none.

e)

f)


8. Each pair of fiugres given below are drawn so that the hypothesis andconclusion are specified. Write the "give" and "to prove" for each p.);

c)

b)

IF YOU HAVE MASTERED THE BEHAVIORAL OBJECTIVES, TAKE YOUR PROGRESSTEST.

ADVANCE STUDY I

1. Construct the triangles determined by each 'triple of measures

listed below. If two triangles may be constructed from each of

the measures construct both. If more than two triangles or no

triangles may be constructed, explain why.

a) m LA = 60, AB = 4, m LB = 90b) m LX = 110, AB = 10, 'BC = 6c) m LW = 18, WU . 3, TU = 2d) MN = 10, NO = 6, 40 = 8e) m LR = 60, RS = 4, m LS = 120f) MN = 3, NO = 4, MO = 8g) AB = 4, BC = 8, m LA = 60h) m LM = 30, m LN . 130, m LO = 20

2. For AXYZ, AXYZ C AYXZ and AXYZ = AZYX, what can you deduce about

AXYZ? Justify your answer.

3. Given AB .1..XY with X - P Y. Points R AND S are on the side of

XY as B, but R and S are on opposite sides of AB. R is on the

same side of All as X. ARBA = ASBA. Prove that LXAR = LYAS.

4. An equivalence relation is defined as a relation among the members

of a set which has the following properties: If a, b, and c are

any members of the set, then

1) a * a (Reflexive)2) If a * b, then b * a (Symmetric)3) If a * b and b * c, then a * c (Transitive)

In applying this relation, you should replace the asterisk by the

relation.

Choose an appropriate set for the FolloNing relations and then

determine which are equivalence relations.

a) is less than

d is a classmate of

b is equal toc is the reciprocal of

e) is taller thanf) is greater than

Is congruence an equivalence relation? Write a supporting

argument to justify your answer.

8

SECTION II

BEHAVIORAL OBJECTIVES

By the completion of the prescribed course of study, you will be

able to:

7. Given any triangle, apply the appropriate definitions,

theorems, and corollaries, to prove statements relating to

isosceles or equilateral triangles.

8. Given any geometric figure, a hypothesis and conclusion, with

an incomplete two-column proof concerning a congruence, com-

plete the proof by using the appropriate definitions, post-

ulates and theorems:

a) Identity Congruence Theorems

b) Definition cf an angle bisector

c) Definition of a bisector of any given angle of a triangle

d) Definition of a median

e) Definition of an isosceles triangle

f) Definition of an equilateral triangle

g) Theorems relating to isosceles and equilateral triangles

h) Definition of a square

i) Definition of a rectangle

j) Congruence of triangles postulates

9. Given a hypothesis and a required conclusion involving a

congruence between overlapping triangles and non-overlapping

triangles, use the appropriate definitions, theorems, and

postulates to write an original proof for this conclusion.

SECTION II

RESOURCES

I. Readings:

1. Moise: #7 pp. 134-136; #8 - #9 pp. 122-125, 132, 134-136,

138-140, 143144.

2. Jurgensen: #7 pp. 207-210; #8 - 19 pp. 201-202, 204.

3. Anderson: #7 pp. 173-175; #8 - #9 pp. 164-166, 177 -179.

4. Lewis: #7 pp. 141-149; #8-09 pp. 126-128, 131132, 136, 139.

5. Nichols: #7 pp. 153-154; #8 - #9 pp. 151, 158.

II. Problems:

1. Moise: #7 pp. 136-137 ex. 1-15; #8 - #9 pp. 125-127 ex. 1-

10, pp. 130-131 ex. 3-14, p. 133 ex. 1-5, 7; pp. 136-137

ex. 1-12, 14, 15, pp. 141-143 ex. 1-6, 9-10, 13, 15, 18-19,

pp. 145-146 ex. 1-11, pp. 147-148 ex. 1-12.

2. Jurgensen: #7 pp. 210-212 ex. 1-26; 18 - #9 pp. 203 ex. 1-

17, pp. 205-207 ex. 1-32.

3. Anderson: #7 pp. 176-177 ex. 1-16; #8 - #9 pp. 162-164 ex.

2-12, pp. 167-178 ex. 1-10, pp. 171-172 ex. 1-14, pp. 180-

181 ex. 1-11.

4. Lewis: #7 pp. 149-152 ex. 1-22; W8 - #9 pp. 129-131 ex. 1-

16, pp. 133-135 ex. 1-16, pp. 137-138 ex. 1-10, pp. 140-

141 ex. 1-12, pp. 158-159 ex. 1-12, pp. 166-167 ex. 1-8, 10,

12.

5. Nichols: #7 p. 169 ex. 1-2; #8 - #9 p. 152 ex. 2-8, p. 157

ex. 2-16, p. 159 ex. 1-3, pp. 165-166 ex. 1-7, p. 169 ex. 3-

7.

510

SELF-EVALUATION II

1. Given WI = 171-6

qa.

Prove: GPM° = LQMN

2. Given aXYZ; MK 1 XZ

FITI 07,

MK= IMJ

Prove: AXYZ is isosceles.

3. For the following statement:

a) Write it in the hypothesisconclusion form.b) Identify the "given" and "to prove".

Draw a figure and write out a proof of the statement:

"The bisector of the vertex angle of an isosceles triangleis perpendicular to the base."

11

. 53


4. Complete the following proof:

Given: D is the midpoint of AT,BA . BC and LA = LC

Prove: LABD = LCBD

PROOF: STATEMENT REASON

1. D is the midpoint of Ar

2. AD . DC

3. BA = BC

4. m LA = m LC

5.

6.

5. Given: AB = DE, FE = C3

lAr, a 1 AU

Prove: LF = LC

1.

2.

3.

4.

5.

6.

6. Given: WX = YX

Z is the midpoint of WY

Prove: XZ is the bisector of LWXY.

12

ADVANCE STUDY II (cunt')

3. b) If Tr is perpendicular to each of three different rays X, XB,

and XC, and XA = XB = XC, prove that AY ... BY = CY.

c) Let L be the edge of.two half-planes H1 and H2. A and B are

two points of L, M is a point in H1, and R is a point in H

such that LMAB ARAB and AM = AR. Prove AMRB is isosceles.

d) Must Wintersect L in (c)?

e) Does tha answer to (c) require that H1 and H2 be coplanar?

15

BIBLIOGRAPHY

1. Moise, Downs: Geometa - (Addison - Wesley

Publishing Co., 1967).

2. Jurgensen, Donnelly, Dolciani: Modern

Geometry (Houghton Mifflin Co., 1965).

3. Anderson, Garon, Gremillion: School Math-

ematics Geometry (Houghton Mifflin Co., J969).

4. Lewis: Geometry 2nd ed. (D. Van Nostrand Co.,

Inc., 1968).

5. Nichols, Palmer, Schacht: Modern Geometry

(Holt, Rinehart and Winston, Inc., 1968).

16

58

Documents

DOCUMENT RESUME ED 069 506 AUTHOR Ninety … · DOCUMENT RESUME. ED 069 506 SE 015 197. AUTHOR Holland, Bill TITLE Learning Activity Package, Geometry. INSTITUTION Ninety Six High